Search

The long-term dynamics of particles involved in an incompressible flow with a small viscosity ($\epsilon>0$) and slow chemical reactions, is depicted by a class of stochastic reaction-diffusion-advection (RDA) equations with a fast advection term of magnitude $1/\epsilon$. It has been shown in [7] the fast advection asymptotics of stochastic RDA equation in $\mathbb{R}^2$ can be characterized through a stochastic partial differential equation (SPDE) on the graph associated with certain Hamiltonian. To simulate such fast advection asymptotics, we introduce and study an asymptotic-preserving (AP) exponential Euler approximation for the multiscale stochastic RDA equation. There are three key ingredients in proving asymptotic-preserving property of the proposed approximation. First, a strong error estimate, which depends on $1/\epsilon$ linearly, is obtained via a variational argument. Second, we prove the consistency of exponential Euler approximations on the fast advection asymptotics between the original problem and the SPDE on graph. Last, a graph weighted space is introduced to quantify the approximation error for SPDE on graph, which avoids the possible singularity near the vertices. Numerical experiments are carried out to support the theoretical results.

It is known that when the diffuse interface thickness $\epsilon$ vanishes, the sharp interface limit of the stochastic reaction-diffusion equation is formally a stochastic geometric flow. To capture and simulate such geometric flow, it is crucial to develop numerical approximations whose error bounds depends on $\frac 1\epsilon$ polynomially. However, due to loss of spectral estimate of the linearized stochastic reaction-diffusion equation, how to get such error bound of numerical approximation has been an open problem. In this paper, we solve this weak error bound problem for stochastic reaction-diffusion equations near sharp interface limit. We first introduce a regularized problem which enjoys the exponential ergodicity. Then we present the regularity analysis of the regularized Kolmogorov and Poisson equations which only depends on $\frac 1{\epsilon}$ polynomially. Furthermore, we establish such weak error bound. This phenomenon could be viewed as a kind of the regularization effect of noise on the numerical approximation of stochastic partial differential equation (SPDE). As a by-product, a central limit theorem of the weak approximation is shown near sharp interface limit. Our method of proof could be extended to a number of other spatial and temporal numerical approximations for semilinear SPDEs., Comment: 58 pages

This paper is concerned with the random effect of the noise dispersion for stochastic logarithmic Schr\"odinger equation emerged from the optical fibre with dispersion management. The well-posedness of the logarithmic Schr\"odinger equation with white noise dispersion is established via the regularization energy approximation and a spatial scaling property. For the small noise case, the effect of the noise dispersion is quantified by the proven large deviation principle under additional regularity assumptions on the initial datum. As an application, we show that for the regularized model, the exit from a neighborhood of the attractor of deterministic equation occurs on a sufficiently large time scale. Furthermore, the exit time and exit point in the small noise case, as well as the effect of large noise dispersion, is also discussed for the stochastic logarithmic Schr\"odinger equation.

We study a class of fully-discrete schemes for the numerical approximation of solutions of stochastic Cahn--Hilliard equations with cubic nonlinearity and driven by additive noise. The spatial (resp. temporal) discretization is performed with a spectral Galerkin method (resp. a tamed exponential Euler method). We consider two situations: space-time white noise in dimension $d=1$ and trace-class noise in dimensions $d=1,2,3$. In both situations, we prove weak error estimates, where the weak order of convergence is twice the strong order of convergence with respect to the spatial and temporal discretization parameters. To prove these results, we show appropriate regularity estimates for solutions of the Kolmogorov equation associated with the stochastic Cahn--Hilliard equation, which have not been established previously and may be of interest in other contexts.

We study the Wasserstein Hamiltonian flow with a common noise on the density manifold of a finite graph. Under the framework of stochastic variational principle, we first develop the formulation of stochastic Wasserstein Hamiltonian flow and show the local existence of a unique solution. We also establish a sufficient condition for the global existence of the solution. Consequently, we obtain the global well-posedness for the nonlinear Schr\"odinger equations with common noise on graph. In addition, using Wong-Zakai approximation of common noise, we prove the existence of the minimizer for an optimal control problem with common noise. We show that its minimizer satisfies the stochastic Wasserstein Hamiltonian flow on graph as well.

This paper focuses on the construction and analysis of explicit numerical methods of high dimensional stochastic nonlinear Schrodinger equations (SNLSEs). We first prove that the classical explicit numerical methods are unstable and suffer from the numerical divergence phenomenon. Then we propose a kind of explicit splitting numerical methods and prove that the structure-preserving splitting strategy is able to enhance the numerical stability. Furthermore, we establish the regularity analysis and strong convergence analysis of the proposed schemes for SNLSEs based on two key ingredients. One ingredient is proving new regularity estimates of SNLSEs by constructing a logarithmic auxiliary functional and exploiting the Bourgain space. Another one is providing a dedicated error decomposition formula and a novel truncated stochastic Gronwall's lemma, which relies on the tail estimates of underlying stochastic processes. In particular, our result answers the strong convergence problem of numerical methods for 2D SNLSEs emerged from [C. Chen, J. Hong and A. Prohl, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016), no. 2, 274-318] and [J. Cui and J. Hong, SIAM J. Numer. Anal. 56 (2018), no. 4, 2045-2069]., Comment: 30 pages

In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with $L^2$-Wasserstein metric tensor, via the Wong--Zakai approximation. We begin our investigation by showing that the stochastic Euler-Lagrange equation, regardless it is deduced from either variational principle or particle dynamics, can be interpreted as the stochastic kinetic Hamiltonian flows in Wasserstein manifold. We further propose a novel variational formulation to derive more general stochastic Wassersetin Hamiltonian flows, and demonstrate that this new formulation is applicable to various systems including the stochastic Schr\"odinger equation, Schr\"odinger equation with random dispersion, and Schr\"odinger bridge problem with common noise., Comment: 34 pages

In this paper, we prove the global existence and uniqueness of the solution of the stochastic logarithmic Schr\"odinger (SlogS) equation driven by additive noise or multiplicative noise. The key ingredient lies on the regularized stochastic logarithmic Schr\"odinger (RSlogS) equation with regularized energy and the strong convergence analysis of the solutions of (RSlogS) equations. In addition, temporal H\"older regularity estimates and uniform estimates in energy space $\mathbb H^1(\mathcal O)$ and weighted Sobolev space $L^2_{\alpha}(\mathcal O)$ of the solutions for both SlogS equation and RSlogS equation are also obtained., Comment: 32 pages

We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic Hamiltonian process in many classical discrete problems, such as the optimal transport problem, Schr\"odinger equation and Schr\"odinger bridge problem (SBP). The stationary and periodic properties of Hamiltonian processes are also investigated in the framework of SBP., Comment: 29 pages

This paper investigates the convergence of density approximations for stochastic heat equation in both uniform convergence topology and total variation distance. The convergence order of the densities in uniform convergence topology is shown to be exactly $1/2$ in the nonlinear case and nearly $1$ in the linear case. This result implies that the distributions of the approximations always converge to the distribution of the origin equation in total variation distance. As far as we know, this is the first result on the convergence of density approximations to the stochastic partial differential equation.

In this article, we consider the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise with diffusion coefficient of sublinear growth. By introducing the spectral Galerkin method, we first obtain the well-posedness of the approximated equation in finite dimension. Then with the help of the semigroup theory and the factorization method, the approximation processes is shown to possess many desirable properties. Further, we show that the approximation process is strongly convergent in certain Banach space via the interpolation inequality and variational approach. Finally, the global existence and regularity estimate of the unique solution process are proven by means of the strong convergence of the approximation process., Comment: 23 pages

In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. To deal with the non-globally monotone coefficient in the considered equation, we first present the exponential integrability properties of the exact and numerical solutions. Then we show the existence and smoothness of the density function of the numerical solution by proving its uniform non-degeneracy in Malliavin sense. In order to analyze the approximate error between the density function of the exact solution and that of the numerical solution, we derive the optimal strong convergence rate in every Malliavin--Sobolev norm of the numerical scheme via Malliavin calculus. Combining the approximation result of Donsker's delta function and the smoothness of the density functions, we prove that the convergence rate in density coincides with the optimal strong convergence rate of the numerical scheme., Comment: 39 pages

Strong and weak approximation errors of a spatial finite element method are analyzed for stochastic partial differential equations(SPDEs) with one-sided Lipschitz coefficients, including the stochastic Allen--Cahn equation, driven by additive noise. In order to give the strong convergence rate of the finite element method, we present an appropriate decomposition and some a priori estimates of the discrete stochastic convolution. To the best of our knowledge, there has been no essentially sharp weak convergence rate of spatial approximation for parabolic SPDEs with non-globally Lipschitz coefficients. To investigate the weak error, we first regularize the original equation by the splitting technique and derive the regularity estimates of the corresponding regularized Kolmogorov equation. Meanwhile, we present the regularity estimate in Malliavin sense and the refined estimate of the finite element method. Combining the regularity estimates of regularized Kolmogorov equation with Malliavin integration by parts formula, the weak convergence rate is shown to be twice the strong convergence rate., Comment: 26pages

This article analyzes an explicit temporal splitting numerical scheme for the stochastic Allen-Cahn equation driven by additive noise, in a bounded spatial domain with smooth boundary in dimension $d\le 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space-time white noise, we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity, we then prove that under very mild assumptions on the initial data, this scheme achieves the optimal strong convergence rate $\OO(\delta t^{\frac 14})$. When $d\le 3$ and the driving noise possesses some regularity in space, we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\OO(\delta t)$., Comment: 33 pages

In this paper, we consider the well-posedness of the weakly damped stochastic nonlinear Schr\"odinger(NLS) equation driven by multiplicative noise. First, we show the global existence of the unique solution for the damped stochastic NLS equation in critical case. Meanwhile, the exponential integrability of the solution is proved, which implies the continuous dependence on the initial data. Then, we analyze the effect of the damped term and noise on the blow-up phenomenon. By modifying the associated energy, momentum and variance identity, we deduce a sharp blow-up condition for damped stochastic NLS equation in supercritical case. Moreover, we show that when the damped effect is large enough, the damped effect can prevent the blow-up of the solution with high probability., Comment: 19 pages

We prove the optimal strong convergence rate of a fully discrete scheme, based on a splitting approach, for a stochastic nonlinear Schr\"odinger (NLS) equation. The main novelty of our method lies on the uniform a priori estimate and exponential integrability of a sequence of splitting processes which are used to approximate the solution of the stochastic NLS equation. We show that the splitting processes converge to the solution with strong order $1/2$. Then we use the Crank--Nicolson scheme to temporally discretize the splitting process and get the temporal splitting scheme which also possesses strong order $1/2$. To obtain a full discretization, we apply this splitting Crank--Nicolson scheme to the spatially discrete equation which is achieved through the spectral Galerkin approximation. Furthermore, we establish the convergence of this fully discrete scheme with optimal strong convergence rate $\mathcal{O}(N^{-2}+\tau^\frac12)$, where $N$ denotes the dimension of the approximate space and $\tau$ denotes the time step size. To the best of our knowledge, this is the first result about strong convergence rates of temporally numerical approximations and fully discrete schemes for stochastic NLS equations, or even for stochastic partial differential equations (SPDEs) with non-monotone coefficients. Numerical experiments verify our theoretical result., Comment: 39 pages

In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schr\"odinger equations driven by a multiplicative $Q$-Wiener process. Beyond the uniform boundedness of moments for high order derivatives of the exact solution, the key requirement of our approach is the exponential integrability of both the exact and numerical solutions. By constructing and analyzing a Lyapunov functional and its discrete correspondence, we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution, which immediately yields the well-posedness of both the continuous and discrete problems. The latter exponential integrability is obtained through a variant of a criterion given by [Cox, Hutzenthaler and Jentzen, arXiv:1309.5595]. As a by-product of this exponential integrability, we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation-type result on the dependence of the noise with first order strong convergence rate., Comment: 26 pages